Problem 19 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. Fourteen added to twice a number.

Step-by-Step Solution

Answer

Answer: 2x + 14

Step 1: Assign a variable for the unknown number

Let x represent the unknown number.

Step 2: Apply the "twice" operation

Multiply x by 2, so we have 2x to represent twice the unknown number.

Step 3: Apply the "added" operation

Add 14 to our expression 2x, resulting in the final mathematical expression: 2x + 14

Key Concepts

Algebraic Expressions

Algebraic expressions are the combination of numbers, variables, and mathematical operations that represent a specific value. Unlike equations, algebraic expressions do not have an equality sign. They are used to describe patterns, relationships, or situations in mathematical terms without resolving to a single solution. For example, to translate the phrase 'twice a number', we use a variable, like 'x', to represent the number, and then multiply it by 2 to indicate doubling it; thus, the expression is '2x'.

• An expression can be as simple as a single term, like '7x', or more complex with several terms, such as '3x + 5 - 2y'.
• Key operations in algebra include addition, subtraction, multiplication, and division, along with exponents and roots.
• When translating phrases, look out for keywords like 'sum' (addition), 'difference' (subtraction), 'product' (multiplication), and 'quotient' (division) to know which mathematical operation to apply.

Equations

An equation is a statement that asserts the equality of two expressions. Equations often include an '=' sign and are central in finding the values of unknown variables. They are the foundation of most problem-solving strategies in algebra. When you come across a phrase like 'the sum of a number and ten is twelve', you translate that into an equation, such as 'x + 10 = 12'. This particular equation can then be solved for 'x' by using algebraic methods to isolate the variable.

Equations take various forms:

Linear Equations

Typically in the form 'ax + b = c', where 'a', 'b', and 'c' are constants, and 'x' is the variable.

Quadratic Equations

These have the standard form 'ax^2 + bx + c = 0', with 'a', 'b', and 'c' as constants, and 'x' representing the variable whose square is involved.

Mathematical Operations

Mathematical operations are the various procedures or actions that produce a new value from one or more input values. Basic mathematical operations include addition, subtraction, multiplication, and division. When solving algebraic expressions and equations, understanding these operations and the order in which they are applied—known as the order of operations—is crucial.

• Addition (+): Combining quantities to find their total.
• Subtraction (-): Determining the difference between quantities.
• Multiplication (\( \times \)): Finding the total when one number is taken times another number.
• Division (\( \div \)): Determining how many times one number is contained within another.

The phrase 'fourteen added to twice a number' would be translated to '2x + 14' by using these operations: multiplication to find 'twice a number' and then addition to add fourteen.